I started with this thought half-jokingly, but now I think I'm serious:
Every single SS# needs to be made public, along with the assigned name. Not just known-breached, but somewhere where it's obvious that anyone could look it up as easily as looking up a phone number or mailing address. Somewhere so public that everyone else knows that they're no longer "secret."
Using Social Security numbers as some sort of proof you're who you say you are is batshit crazy these days. But we all pretend that it's still secure somehow. If there was an embarrassingly public list of all SS#s, then banks would be forced to improve their vetting of applicants.
At the same time, the onus needs to be on a financial institution to prove that I opened an account. If I discover a line of credit in my name, all I should need to do is disavow it, and make the lender prove that I was the one who authorized it.
Yes, this will increase the cost of doing business. But that increased cost is already here, just born randomly and disproportionately by the victims of "identity theft."
Want to issue a credit card with a $15,000 limit? Have the applicant walk into a branch, provide a thumbprint take their picture. Or get them on a video call standing in front of their house, attesting that they're the person they say they are. Or if you think that's too inconvenient you can take on all the risk if the borrow later disavows the debt.
A piece of paper or online form with the magic numbers is just not enough.